Arranging a special permutation of the sequence (1,2,3..., n) into a circle

Question

Given the first n natural numbers (1,2,3,...,n). We arrange a permutation of these numbers into a circle, such that the sum of any two adjacent numbers is prime. I have proven that there is at least one solution for n=4; n=6; n=8; n=10; I have tried but found no solutions for n=3; n=5; n=7; n=9. My Question: Is it true that there is always a solution for any n = even, but no solution for n = odd??. Here are my two solutions for n=10: (1,2,5,6,7,10,9,8,3,4); and (1,4,3,8,5,2,9,10,7,6). Notice that each of the previous two permutations represent a Hamiltonian cycle (a closed circuit) through the ten vertices (1,2,3,...,10)

Answer 1

I have manually determined a solution to the problem for n=60; The Hamiltonian cycle goes through all 60 vertices and is closed. Due to lack of place, I had to fold the cycle a few times. Note: this is one solution of many.